Mathematical Modeling of Stem Cells Related to Cancer

In this chapter we present an overview of different mathematical and numerical approaches to describe stem cell proliferation and differentiation and the development of small cancer stem cell populations that are origins of neoplasm disease. The purpose of this chapter is not to scare the reader with complex mathematical and numerical analysis. Instead we aim to summarize the wide range of possibilities that mathematics and computer sciences can offer to experimentalists and theoretical biologists. We briefly introduce recently developed models that address different aspects of stem cell dynamics and cancer development. First we focus on models of periodic hematological diseases having origin in abnormal behavior of hematopoietic stem cells in bone marrow. We also present models, less organ specific, describing the differentiation and possible mutation of stem cells which can give rise to cancer. We introduce a simple Monika Joanna Piotrowska, Heiko Enderling, Uwe an der Heiden et al. 2 model which simulates the crucial interplay of proliferating and quiescent stem cells, and we discuss its potential application to treatment design. One of the presented models illustrates in detail the role of nuclear factor κB-complex in proliferation of adult neural stem cells. Finally the reader will become acquainted with a computational model approach able to simulate stem cell dependent tissue development, homeostasis, and recovery from external perturbations.